Games of chance can be thought of as coming in three basic varieties. Games in which there are no player decisions, and the result is essentially entirely random; games where the player makes decisions to the extent that the player chooses among different types of wagers; and games where the player makes decisions that affect the outcome of the game.
An example of the first type of game is a standard three-reel spinning slot machine. The player makes a wager, but provides no other input. The results of the game are shown to the player in the form of indicia on the reels, and the player receives an award in the case of a winning result. This type of game can be found, for example, in machines that spin mechanical reels or that simulate the reels on a video display, which have been adapted for casino or other gambling environments, as well as on a home computer or game console.
The second type of game of chance noted above provides different ways to place bets, or different types of bets on a single game. Each type of bet carries its own set of rules, and its own payoff schedule and odds of winning. Some bets may provide better expected return than others, but other than deciding which bet to make on a particular game (which may affect expected return), the decisions made by the player in this second type of game again have no effect on the result of winning or losing. There are many examples of this second type of game of chance, as for instance, gaming machines and casino table games including craps, roulette, keno and Baccarat, all of which may be played with live dealers in a casino, on a slot machine or on a home computer or game console.
The third variety of games of chance considered herein involves decisions that are made by the player that have a direct impact on the result of the game. Games of this nature include BlackJack, Pai Gow Poker, Caribbean Stud Poker, Let it Ride and Video Poker, among others. In each of these games, the player receives an initial hand and then makes one or more decisions about how to proceed in the game. The player's decision-making in these games has a causal effect on the outcome. Specifically, the player may wish to try to make these decisions using the best odds from tables and strategies known to the player, or may play a hunch about streaks being observed, or make a decision under some influence or factor (e.g., fear of jeopardizing a large bet, or to take advantage of the history of the table, such as is done by a “card counting” blackjack player). Of course, a “decision” could also be an unintended mistake, causing a worse expected result. This third type of game is thus to be contrasted to the first and second types where the player's decisions do not affect the winning or losing outcome of the game.
In this third variety of game, the designer of the game will typically do a mathematical analysis of all possible starting hands (using a card game format for example), and all possible outcomes after each possible decision. For any combination of game rules and pay schedule, there is an optimal payout percentage that is computed. This optimal payout percentage is the percentage of a given wager that would be returned to a player that made the optimal decision on every hand over the long run. In the case of a game of chance used for gambling, this optimal payout percentage could be thought of as the worst-case payout percentage for the casino. That is, the percentage of wagers that will be returned to the very best players over the long run. The concept of optimal payout percentage is governed by the laws of probability and statistics, and is well known by those familiar with the art.
Most games of chance that are used for casino wagering have an optimal payout percentage set at less than 100%. This percentage is returned to the player and the balance (between the optimal percentage and 100%, sometimes called the “house edge”) is retained by the casino as a profit.
In real life, most games will pay back less than their optimal percentage. This occurs because players often make non-optimal decisions when playing. There are many reasons for players to make non-optimal decisions, such as the game is one for which the player does not understand the optimum strategy, or mistakes and oversights are made by the player, including making non-optimum moves for other reasons such as hunches or superstitions. In the long run, this non-optimal play will result in a greater profit for the casino beyond the house edge.
Because of the highly competitive nature of casino gambling, this greater profit has allowed casinos to offer games with a very high optimal return percentage, knowing that, through mistakes and other non-optimal play, they will receive a better profit than the mathematical house edge. Specifically, it is common to find Blackjack (also known as “21”) games with optimal returns of over 98%, and video poker games with optimal returns over 99%. For example, it is well known that a “Jacks or Better” video draw poker with a “9-6” paytable has a return of about 99.54%. (Note that a 9-6 paytable refers to a full house payout of 9 for 1 and a flush payout of 6 for 1.) Most “Jacks or Better” draw poker games have the same paytable at all values except Flush and Full House, and these values are modified to adjust the optimal payout percentage. Table A shows a 9-6 Jacks or Better Paytable for a 1 coin wager.
TABLE ARoyal Flush800Straight Flush50Four of a Kind25Full House9Flush6Straight4Three of a Kind3Two Pair2Pair of Jacks or Better1Competition can be so strong in certain areas for certain customers that it is not uncommon to find machines that offer optimal payouts of over 100%, with the knowledge that these machines will still be profitable as a result of non-optimal play. Well-known examples of this are “Full Pay Deuces Wild” and 9-7 or 10-6 “Jacks or Better” video poker. The paytable for a Full Pay Deuces Wild which has an optimal payout of about 100.76% is shown in Table B.
TABLE BRoyal Flush800Four Deuces200Royal Flush w/deuces25Five of a Kind15Straight Flush9Four of a Kind5Full House3Flush2Straight2Three of a Kind1As a result of advertising and word of mouth between players, it is well known that there are casino games that offer an opportunity to play the games with little or no house advantage, if they learn to play the optimum strategy. This is a very attractive proposition for certain players, because there are additional benefits offered to the prospect of breaking even while playing the game. Casinos have “slot clubs” which are akin to “frequent flyer” programs, but for slot machine players. The casino monitors play through the use of a “player tracking card,” and typically returns between 0.5 and 3% of the player's play in the form of cash back and “comps”. Comps can be anything of value, and are typically discounted or free rooms in the hotel, discounted or free food and entertainment. Additionally, there is the attraction of free drinks at many casinos, and the ambiance, excitement and general entertainment provided by playing games of chance in a casino environment. These benefits provided to attract gamblers, combined with optimal play returns of over 99%, often make the labor of learning optimum play a worthwhile endeavor for many players.
There have been many books written, and lately computer simulations written, that teach players optimum strategy. The computer simulations, among other features allow you to play the game as if you were in a casino, and alert the player that a non-optimum choice was made. In addition, the simulations may provide other features, such as tracking the overall quality of play, and showing the player the accuracy and/or expected loss as a result of a move or a mistake made (if any). The purpose of such a simulation is to learn through repetition and memorization which decisions to make for which types of hands in the game.
It should be noted that in all of these games where the player makes decisions, the optimal strategy is one based on the expected value of one or more random events. That is, the best choice is the one that over the long run is expected to produce the best results. Because there is information about the random event(s) that is unknown at the time of a given decision, there will be times that a different choice would generate a better result. For instance, where optimum Blackjack strategy dictates hitting a 16 when the dealer shows 7 or higher, if the “hit” is a 10 and the dealer's hole card was a 5, then in that particular case the player could have won the hand by standing (in which case the dealer would have “busted”). That information—the hole card as well as the player's next card (the top card on the deck)—was unknown to the player at the time a decision was to be made.